(i) We choose \(N\in {\mathbb {N}_{odd}}\)  so large, such that we have the assumed sign of the the one element in \(\left(\Phi _k^{\prime }(0)\right)_{{k}\le N}\) . The restriction of \(J\)  to the \(\frac{N+1}{2}\) -dimensional space \({\mathcal {D}(J^{(N)})}\)  preserves coercivity. The continuity of \(J^{(N)}\)  therefore guarantees the existence of a minimizer \(\hat{\alpha }^{(N)}\in {\mathcal {D}(J^{(N)})}\) . As before we see that \(J(\hat{\alpha }^{(N)})=\inf J^{(N)}<0\) , so in particular \(\hat{\alpha }^{(N)}\ne 0\) . Next we observe that \({\mathcal {D}(J^{(N)})}\subset {\mathcal {D}(J)}\) , i.e., \(J(\hat{\alpha }^{(N)})\ge \inf J= J(\hat{\beta })\)  for a minimizer \(\hat{\beta }\in {\mathcal {D}(J)}\)  of \(J\) . Let us define \(\hat{\beta }^{(N)}_k=\hat{\beta }_k\)  for \({k}\le N\)  and \(\hat{\beta }^{(N)}_k=0\) . Since the Fourier-series \(\beta (t) = \sum _k \hat{\beta }_k e_k(t)\)  converges in \(L^4(\) , cf. Theorem 4.1.8 in [1]}, we see that \(\hat{\beta }^{(N)}\rightarrow \hat{\beta }\)  in \({\mathcal {D}(J)}\) . By the minimality of \(\hat{\alpha }^{(N)}\in {\mathcal {D}(J^{(N)})}\)  and continuity of \(J\)  we conclude
\(\inf _{{\mathcal {D}(J)}}J\le J(\hat{\alpha }^{(N)})\le J(\hat{\beta }^{(N)})\longrightarrow J(\hat{\beta })=\inf _{{\mathcal {D}(J)}}J.\) 

Hence \(\lim _{N\rightarrow \infty } J(\hat{\alpha }^{(N)})=\inf J\)  as claimed.
(ii) Since \({\mathcal {D}(J^{(N)})}\subset {\mathcal {D}(J^{(N+1)})}\subset {\mathcal {D}(J)}\)  we see that \(J(\hat{\alpha }^{(N)})\ge J(\hat{\alpha }^{(N+1)})\ge \inf J\)  so that in particular the sequence \((J(\hat{\alpha }^{(N)}))_N\)  is bounded. By coercivity of \(J\)  we conclude that \((\hat{\alpha }^{(N)})_N\)  is bounded in \({\mathcal {D}(J)}\)  so that there is \(\hat{\alpha }\in {\mathcal {D}(J)}\)  and a subsequence (again denoted by \((\hat{\alpha }^{(N)})_N\) ) such that
\(\hat{\alpha }^{(N)}\rightharpoonup \hat{\alpha } \qquad \text{ in }~{\mathcal {D}(J)}.\) 

By part (i) and weak lower semi-continuity of \(J\)  we obtain
\(\inf J=\lim _{N\rightarrow \infty }J(\hat{\alpha }^{(N)})\ge J(\hat{\alpha }),\) 

i.e., \(\hat{\alpha }\)  is a minimizer of \(J\) . Recall that \(J(\cdot )= \frac{1}{4}{\interleave \cdot \interleave }^4+J_1(\cdot )\)  where \(J_1\)  is weakly continuous, cf. proof of Theorem REF . Therefore, since \(\hat{\alpha }^{(N)}\rightharpoonup \hat{\alpha }\)  and \(J(\hat{\alpha }^{(N)})\rightarrow J(\hat{\alpha })\)  we see that \({\interleave \hat{\alpha }^{(N)}\interleave }\rightarrow {\interleave \hat{\alpha }\interleave }\)  as \(N\rightarrow \infty \) . Since \({\mathcal {D}(J)}\)  is strictly uniformly convex, we obtain the norm-convergence of \((\hat{\alpha }^{(N)})_N\)  to \(\hat{\alpha }\) .
